This is going to be difficult.
Usually a J-1 non-immigrant exchange student has a two year home country residency requirement upon termination of his/her program in the United States. Please check with your FSA to see if they can assist with your adjustment application as you may need a J-1 waiver of the two year requirement before USCIS/State/USAID will allow adjustment. The adjustment is usually based on hardship if the alien is returned to his home country.
If your FSA is unable ot help you, please obtain local immigration counsel at once.
Here's a link to USCIS's web page explaining the two year home country residency requirement and the waiver procedure.
First determine whether or not the J visa subjects your boyfriend to the two-year foreign residency requirement. On the visa itself (in his passport), there will be a notation "subject (or not) to 212(e) two-year foreign residency requirement.
Second, if he is not subject, you can file. He can work without authorization as a marriage to a USC and a granted adjustment application will cure the unauthorized employment. See INA 245(c). However, if his employer knows he is working without authorization, the employer is subject to sanctions. When filing for his permanent residency, he can also apply for employment authorization.
If he is subject to the two year foreign residency requirement, there are two avenues to deal with the issue.
[Disclaimer - This is answer is not to be construed as legal advice. Prior to filing with USCIS, you should contact and consult with an immigration attorney.]